An implicit finite-element method for high-speed flows

A fast algorithm is presented for constructing continuous lines, made up of element sides, which pass once through each node of a general unstructured triangular mesh and which are generally aligned in prescribed directions. The lines are used as the basis of an adaptive fully implicit unstructured grid procedure for the solution of two-dimensional problems of steady compressible inviscid and laminar viscous high-speed flows, where the equation system is solved by line relaxation using a block tridiagonal equation solver. For three-dimensional laminar viscous simulations it is proposed to utilize an implicit/explicit finite-element formulation. In the vicinity of solid walls a grid exhibiting structure in the normal direction is employed while, away from this region, the grid will be totally unstructured. In the structured region, lines in the normal direction to the wall are readily identified, while lines in the surfaces parallel to the solid wall are constructed using the proposed two-dimensional procedure. The implicit algorithm is then used in the structured region and the equation solution is achieved via line relaxation. An explicit form of the solution algorithm is used elsewhere. To illustrate the performance of the proposed method, solutions are obtained for both transonic inviscid and transonic and hypersonic laminar viscous problems in two dimensions. The application of the proposed procedure to the solution of three-dimensional hypersonic laminar viscous flow over a double ellipsoid configuration is also described.     

Flow and conjugate heat exchange in a rotating cavity with an axially supplied working medium

Numerical modeling of inviscid compressible gas flow past a linear cascade of airfoils executing small harmonic translational and rotational vibrations is carried out. The control volume method on moving unstructured grid structures is used for discretization of basic equations. The influence of the frequency and phase characteristics of the airfoils on the pressure distribution over their surface and the lift coefficient is investigated. The results of numerical calculations are compared to the available calculated and experimental data.     

基于非结构自适应网格技术的高超声速流动数值模拟

发展了一种基于非结构网格的自适应方法,对高超声速无粘流场进行了数值模拟。根据流场参数的变化梯度确定加密边,由加密准则进行自适应网格剖分后得到分布合理的较密网格。通过预先生成的初始极密表面网格将边界的加密点投影到边界上,使得边界保持初始外形。通过求解三维Euler 方程,对三维双椭球高超声速绕流问题进行了数值模拟,计算结果和实验数据相吻合,表明了该文所建立方法的正确性和可靠性。     

An upwind kinetic flux vector splitting method on general mesh topologies

An upwind flux vector splitting algorithm which utilizes the moments of the Boltzmann equation to derive the Euler equations for inviscid compressible flow has been used with a variety of grid types. Although the upwind approach offers the potential for accurate flow simulations, it is necessary to ensure that such procedures can be utilized on realistic grids. In this paper, an upwind algorithm is used with structured multiblock grids, unstructured grids of triangles and hybrid structured/unstructured grids to solve realistic compressible flow problems in two dimensions.     

Flow feature aligned grid adaptation

A flow feature aligned grid adaptation method is proposed for the solution of Euler and Navier–Stokes equations for compressible flows, motivated by the desire for an efficient grid system for an accurate and robust solution method to best resolve flow features of interest. The method includes extraction of the flow features; generation of the embedded flow feature aligned structured blocks combined with unstructured grid generation for the rest of the flowfield; and adaptation of the hybrid grid for high flow feature resolution. The feature alignment makes it possible to maintain the high resolution property for both shock waves and shear layers of the approximate Riemann solvers and the higher order reconstruction schemes based on one‐dimensional derivation and dimensional splitting. High grid efficiency is obtained with highly anisotropic directional grid corresponding to the feature directions. The computational procedure is described in details in the paper and its application to flow solutions involving shock waves, boundary layers, wakes and shock boundary layer interaction are demonstrated. Its accuracy, efficiency and robustness are discussed in comparison with an anisotropic unstructured grid adaptations for the shock boundary layer interaction case. Copyright © 2006 John Wiley & Sons, Ltd.     

Heat transfer in flow of an incompressible viscous liquid between disks

Using approximate equations of motion, an investigation has been made of the development of steady laminai radial flow of a viscous incompressible liquid in the gap between parallel disks. In the region of hydrodynamically stable flow the heat transfer problem is solved for a given constant heat flux at the wall.     

The inverse problem in creeping film flows

We consider the inverse problem for gravity-driven free surface flows at vanishing Reynolds numbers. In contrast to the direct problem, where information about the underlying topographic structure is given and the steady free surface shape and the flow field are unknown, the inverse problem deals with the flow along unknown topographies. The bottom shape and the corresponding flow field are reconstructed from information at the steady free surface only. We discuss two different configurations for the inverse problem. In the first case, we assume a given free surface shape, and by simplifying the field equations, we find an analytical solution for the corresponding bottom topography, velocity field, wall shear stress, and pressure distribution. The analytical results are successfully compared with experimental data from the literature and with numerical data of the Navier–Stokes equations. In the second inverse problem, we prescribe a free surface velocity and then solve numerically for the full flow domain, i.e. the free surface shape, the topography and simultaneously the wall shear stress and the pressure field. The results are validated with the numerical solution of the corresponding direct problem.     

Viscous-inviscid interaction schemes for external aerodynamics

For aerofoils a calculation, which involves the coupling of the external inviscid flow with the viscous flow in the boundary layer and the wake, still provides a worthwhile alternative to the solution of the ‘time-averaged’ Navier-Stokes equations. Classical viscous-inviscid interaction methods which can be extended to include flows with separations and significant pressure gradients across the boundary layer are described. Basic theoretical principles of interactive methods in two dimensions are discussed. The extension of the classical methods leads to generalisations of the concept of displacement thickness and the momentum integral equation. The boundary conditions for the equivalent inviscid flow (EIF) are also described and these also include the effect of normal pressure gradients. An integral method based on the lag-entrainment method for the calculation of the turbulent boundary layer is described. The correlations associated with the method are extended to include separated flow. Two methods of solving the boundary-layer equations through a separation region are described: the inverse method and the quasi-simultaneous method. Principles of techniques for coupling the flows are described and the properties of the direct, fully inverse, semi-inverse and quasi-simultaneous methods are discussed. Results from a method for incompressible flow about a stalled aerofoil, a method for compressible flow about a high-lift aerofoil and a method for compressible flow about a transonic aerofoil are compared with experimental results. The current situation regarding the development of viscous-inviscid interaction methods is briefly summarized and future possibilities are considered.     

A parallel, implicit, multi-dimensional upwind, residual distribution method for the Navier-Stokes equations on unstructured grids

A multi-dimensional cell-vertex upwind discretization technique for the Navier-Strokes equations on unstructured grids is presented. The grids are composed of linear triangles in two and linear tetrahedra in three space dimensions. The nonlinear upwind schemes for the inviscid part can be viewed as a multi-dimensional generalization of the Roe-scheme, but also as a special class of Petrov-Galerkin schemes. They share with these schemes a compact Galerkin stencil, and are in addition monotonic by construction. The Petrov-Galerkin interpretation of the discretization technique allows a straightforward extension to the Navier-Strokes equations. For linear elements this boils down to a Galerkin discretization for the viscous terms. Compared to standard finite-volume methods on these grids, the method shows an increased accuracy, which becomes comparable with structured grid algorithms. The spatially discretized set of equations is integrated in time using the Backward Euler time integration method. The full Jacobian matrix is computed, either numerically by finite differences or approximated analytically, and stored. The resulting set of linear equations is solved by a Block MILU(0) preconditioned Krylov subspace method. For this purpose the Aztec library of SANDIA National Laboratories is used, which also takes care of the parallelization process and completely hides the details for the user. Results are presented for a two-dimensional turbulent shock wave boundary layer interaction in a nozzle and the turbulent flow over an ogive cylinder. All computations have been performed on the Cray T3E of the Technical University of Delft.     

Calculation of forces and moments in vortex methods

An alternative formulation for the calculation of forces and moments acting on a body in a three-dimensional unsteady viscous incompressible flow field is derived. The formulation is especially useful when the Navier-Stokes equations are solved in vorticity formulation.     

An efficient parallel computation of unsteady incompressible viscous flow with elastic moving and compliant boundaries on unstructured grids

This paper presents the development and validation of a parallel unstructured‐grid fluid–structure interaction (FSI) solver for the simulation of unsteady incompressible viscous flow with long elastic moving and compliant boundaries. The Navier–Stokes solver on unstructured moving grid using the arbitrary Lagrangian Eulerian formulation is based on the artificial compressibility approach and a high‐order characteristics‐based finite‐volume scheme. Both unsteady flow and FSI are calculated with a matrix‐free implicit dual time‐stepping scheme. A membrane model has been formulated to study fluid flow in a channel with an elastic membrane wall and their interactions. This model can be employed to calculate arbitrary wall movement and variable tension along the membrane, together with a dynamic mesh method for large deformation of the flow field. The parallelization of the fluid–structure solver is achieved using the single program multiple data programming paradigm and message passing interface for communication of data. The parallel solver is used to simulate fluid flow in a two‐dimensional channel with and without moving membrane for validation and performance evaluation purposes. The speedups and parallel efficiencies obtained by this method are excellent, using up to 16 processors on a SGI Origin 2000 parallel computer. A maximum speedup of 23.14 could be achieved on 16 processors taking advantage of an improved handling of the membrane solver. The parallel results obtained are compared with those using serial code and they are found to be identical. Copyright © 2005 John Wiley & Sons, Ltd.     

Computation of transonic viscous flow over airfoils with control by an elastic membrane and comparison with control by passive ventilation

Summary The performance of transonic wings can be influenced by control of the shock/boundary layer interaction (SBLI) using an adaptive surface in the shock region. This is achieved by inserting a cavity into the airfoil and covering it with an elastic membrane. The theoretical methods for the computation of transonic viscous flow around airfoils with control are presented. The zonal solution method consists of numerical and analytical parts. The influence of the adaptive wall on the flow field over a modern transonic airfoil has been investigated. The flow parameters have been varied up to off-design conditions to understand the physical effects of the control. Previous investigations of a second way of control, the passive ventilation, allow a comparison of these two methods.     

An implicit–explicit solution method for electro‐osmotic flow through three‐dimensional micro‐channels

This paper presents a comprehensive finite‐element modelling approach to electro‐osmotic flows on unstructured meshes. The non‐linear equation governing the electric potential is solved using an iterative algorithm. The employed algorithm is based on a preconditioned GMRES scheme. The linear Laplace equation governing the external electric potential is solved using a standard pre‐conditioned conjugate gradient solver. The coupled fluid dynamics equations are solved using a fractional step‐based, fully explicit, artificial compressibility scheme. This combination of an implicit approach to the electric potential equations and an explicit discretization to the Navier–Stokes equations is one of the best ways of solving the coupled equations in a memory‐efficient manner. The local time‐stepping approach used in the solution of the fluid flow equations accelerates the solution to a steady state faster than by using a global time‐stepping approach. The fully explicit form and the fractional stages of the fluid dynamics equations make the system memory efficient and free of pressure instability. In addition to these advantages, the proposed method is suitable for use on both structured and unstructured meshes with a highly non‐uniform distribution of element sizes. The accuracy of the proposed procedure is demonstrated by solving a basic micro‐channel flow problem and comparing the results against an analytical solution. The comparisons show excellent agreement between the numerical and analytical data. In addition to the benchmark solution, we have also presented results for flow through a fully three‐dimensional rectangular channel to further demonstrate the application of the presented method. Copyright © 2007 John Wiley & Sons, Ltd.     

Slip flow and heat transfer in rectangular and circular microchannels using hybrid FE/FV method

Rarefied gas flows typically encountered in MEMS systems are numerically investigated in this study. Fluid flow and heat transfer in rectangular and circular microchannels within the slip flow regime are studied in detail by our recently developed implicit, incompressible, hybrid (finite element/finite volume) flow solver. The hybrid flow solver methodology is based on the pressure correction or projection method, which involves a fractional step approach to obtain an intermediate velocity field by solving the original momentum equations with the matrix‐free, implicit, cell‐centered finite volume method. The Poisson equation resulting from the fractional step approach is then solved by node based Galerkin finite element method for an auxiliary variable, which is closely related to pressure and is used to update the velocity field and pressure field. The hybrid flow solver has been extended for applications in MEMS by incorporating first order slip flow boundary conditions. Extended inlet boundary conditions are used for rectangular microchannels, whereas classical inlet boundary conditions are used for circular microchannels to emphasize on the entrance region singularity. In this study, rarefaction effects characterized by Knudsen number (Kn) in the range of 0 ⩽ Kn ⩽ 0.1 are numerically investigated for rectangular and circular microchannels with constant wall temperature. Extensive validations of our hybrid code are performed with available analytical solutions and experimental data for fully developed velocity profiles, friction factors, and Nusselt numbers. The influence of rarefaction on rectangular microchannels with aspect ratios between 0 and 1 is thoroughly investigated. Friction coefficients are found to be decreasing with increasing Knudsen number for both rectangular and circular microchannels. The reduction in the friction coefficients is more pronounced for rectangular microchannels with smaller aspect ratios. Effects of rarefaction and gas‐wall surface interaction parameter on heat transfer are analyzed for rectangular and circular microchannels. For most engineering applications, heat transfer is decreased with rarefaction. However, for fluids with very large Prandtl numbers, velocity slip dominates the temperature jump resulting in an increase in heat transfer with rarefaction. Depending on the gas‐wall surface interaction properties, extreme reductions in the Nusselt number can occur. Present results confirm the existence of a transition point below and above wherein heat transfer enhancement and reduction can occur. Copyright © 2011 John Wiley & Sons, Ltd.     

Finite element solution of free‐surface ship‐wave problems

An unstructured finite element solver to evaluate the ship‐wave problem is presented. The scheme uses a non‐structured finite element algorithm for the Euler or Navier–Stokes flow as for the free‐surface boundary problem. The incompressible flow equations are solved via a fractional step method whereas the non‐linear free‐surface equation is solved via a reference surface which allows fixed and moving meshes. A new non‐structured stabilized approximation is used to eliminate spurious numerical oscillations of the free surface. Copyright © 1999 John Wiley & Sons, Ltd.     

A survey of flow at low pressures

The theory of internal flows at such low pressures that the Navier-Stokes equations are not valid is reviewed. In the extreme low pressure range (free molecular flow), where the flow rate is independent of pressure, the basic theory is well developed. However, the influence of channel lenght, cross sectional shape, or surface characteristics have not been totally defined. Slip theory, which extends the low pressure range of the Navier-Stokes equation by modifying the wall boundary conditions, is discussed. The theory provides a good basis for experimental data correlation, but is not adequate to extend the viscous equations into the free molecular range. The empirical methods used to provide a smooth transition from slip to free molecular flow are reviewed, and a method of obtaining an equation for the total flow regime is illustrated. The extensive work in physical gas dynamic is discussed. Boltzmann's equation which has been solved numerically for small pressure gradients was found to be valid only in the viscous regime near the free molecular flow range.     

三声道燃油超声波流量计内部流场数值研究

针对三声道燃油超声波流量计圆截面多弯曲管道,基于不可压缩雷诺时均Navier-Stokes方程建立流量计管道内计算流体动力学仿真模型。使用重整化群k-ε湍流模型、有限容积法和结构化网络进行离散化,并在近壁区采用标准的壁面函数法修正,完成对三声道燃油超声波流量计的管道内部湍流的数值模拟。数值研究结果表明,在8种不同进口速度条件下,3个声道的流场沿中心线均是不均匀分布;H-H声道流场变化最大,P-P声道流场最平稳;P-P声道测量精度最好,而H-H声道测量精度最差。     

Finite element multigrid solution of Euler flows past installed aero-engines

A finite element based procedure for the solution of the compressible Euler equations on unstructured tetrahedral grids is described. The spatial discretisation is accomplished by means of an approximate variational formulatin, with the explicit addition of a matrix form of artificial viscosity. The solution is advanced in time by means of an explicit multi-stage time stepping procedure. The method is implemented in terms of an edge based representation for the tetrahedral grid. The solution procedure is accelerated by use of a fully unstructured multigrid algorithm. The approach is applied to the simulation of the flow past an installed aero-engine nacelle, at three different free stream conditions. Comparison is made between the numerical predictions and experimental pressure observations.     

注射模充模流动和传热过程的理论与算法

从粘性流体力学的质量、动量和能量方程出发，针对塑料注射成型特点，基于量纲分析，建立了适合于充模分析的数学模型。控制方程的求解主要包括三个阶段：压力场、温度场和流动前沿位置的确定。数值求解采用有限元法求解压力场、有限差分法求解温度场、控制体积法确定流动前沿位置，并详细讨论了数值算例的分析结果。